ODE
\[ \sqrt {a^2-x^2} \left (x y(x) y''(x)-x y'(x)^2-y(x) y'(x)\right )=b x y'(x)^2 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0748367 (sec), leaf count = 53
\[\left \{\left \{y(x)\to c_2 e^{-\frac {\sqrt {a^2-x^2}}{b}} \left (b \sqrt {a^2-x^2}+c_1\right ){}^{\frac {c_1}{b^2}}\right \}\right \}\]
Maple ✓
cpu = 0.564 (sec), leaf count = 50
\[ \left \{ y \left ( x \right ) ={\it \_C2}\,{{\rm e}^{\int \!{x\sqrt {{a}^{2}-{x}^{2}} \left ( b{a}^{2}-b{x}^{2}-{\it \_C1}\,\sqrt {{a}^{2}-{x}^{2}} \right ) ^{-1}}\,{\rm d}x}} \right \} \] Mathematica raw input
DSolve[Sqrt[a^2 - x^2]*(-(y[x]*y'[x]) - x*y'[x]^2 + x*y[x]*y''[x]) == b*x*y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> ((b*Sqrt[a^2 - x^2] + C[1])^(C[1]/b^2)*C[2])/E^(Sqrt[a^2 - x^2]/b)}}
Maple raw input
dsolve((a^2-x^2)^(1/2)*(x*y(x)*diff(diff(y(x),x),x)-x*diff(y(x),x)^2-y(x)*diff(y(x),x)) = b*x*diff(y(x),x)^2, y(x),'implicit')
Maple raw output
y(x) = _C2*exp(Int(x*(a^2-x^2)^(1/2)/(b*a^2-b*x^2-_C1*(a^2-x^2)^(1/2)),x))